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Radiation from accelerated particles in

relativistic jets with shocks, shear-flow

and reconnections Ken Nishikawa Physics/UAH/NSSTC

XXVII Texas Symposium on Relativistic Astrophysics

Dallas, TX, December 8 - 13, 2013

P. Hardee (Univ. of Alabama, Tuscaloosa)

Y. Mizuno (National Tsing Hua University, Taiwan)

I. Duţan (Institute of Space Science, Rumania)

B. Zhang (Univ. Nevada, Las Vegas)

M. Medvedev (Univ. of Kansas)

A. Meli (Univ. of Gent)

E. J. Choi (KAIST)

K. W. Min (KAIST)

J. Niemiec (Institute of Nuclear Physics PAN)

Å. Nordlund (Niels Bohr Institute)

J. Frederiksen (Niels Bohr Institute)

H. Sol (Meudon Observatory)

M. Pohl (U‐Potsdam/DESY)

D. H. Hartmann (Clemson Univ.)

A. Marscher (Boston Univ.)

J. Gómez (IAA, CSIC)

Outline

1. Magnetic field generation and particle

acceleration in kinetic Kelvin-Helmholtz

instability

2. Self-consistent radiation method using PIC

simulations

3. Synthetic spectra in shocks generated by

the Weibel instability

5. Strong magnetic field amplification with

colliding jets with magnetic fields

6. Acceleration in recollimation shock

7. Summary

8. Future plans

Simulations of Kinetic Kelvin-Helmholtz instability

with counter-streaming flows (γ0 = 3, mi/me=1836)

Ä Ä

ÄÄ

Alves et al. (2012), Grismayer et al. (2013)

magnetic field lines electron density

Simulations of KHI with core and sheath jets

Mizuno, Hardee & Nishikawa, ApJ, 662, 835, 2007

RMHD, no wind ω=0.93, time=60.0

case of Vtheath = 0

γj = 15, mi/me = 20

By

By

New KKHI simulations with core and sheath jets in slab geometry

Nishikawa et al. 2013

eConf C121028

(arXiv:1303.2569) By

Jx

Electric field generation by KKHI with real mass ratio

γj = 15, mi/me = 1836, t = 30/ωpe

Ez

Ez By

t = 70/ωpe

(Nishikawa et al. Annales Geophysicae, 2013)

Motion of the electrons across the shear surface produce electric

currents which generate magnetic field

t = 30/ωpe

Jx

By

γj = 1.5, mi/me = 20

By in the x – z plane γ = 15

mi/me = 1386 tωpe = 70 mi/me = 1 tωpe = 200

KKHI with electron-positron jet (core-sheath scheme)

(Nishikawa et al. in preparation, 2013c)

t = 200/ωpe

By

By

Bz

γ= 15

Generation of magnetic field in electron-positron jet-sheath plasmas

By t = 200/ωpe By

(Nishikawa et al. in preparation, 2013c)

no DC magnetic field is found

at this time Jx with arrows (By, Bz)

(Liang et al. 2013)

Magnetic Field Generation and Particle Energization at Relativistic

Shear Boundaries in Collisionless Electron-Positron Plasmas (2D simulation)

p0 = px/mc tωe =1000 3000

p0 =

5

30

60

Bx

Study of the relativistic velocity shear interface KKHI instability

– 4 –

Note that eq.(17) is ident ical to eq.(14) and contains no magnet ic field terms. Thus, the unstable

electrostat ic solut ion associated with a equal density counter-streams on either side of a shear

surface is ident ical to the unstable electrostat ic solut ion associated with interpenetrat ing equal

density beams and is just the classic electrostat ic two-stream instability.

The transverse electric field components, i.e., t ransverse to the wavevector, magnet ic field and

streaming direct ion, had the following dispersion relat ion (eq. 3.34 in the dissertat ion)

−ω2 + k2c2 −γ2ω2

p

2

(ω− kV )2

− (kV − ω)2 + Ω2γ

+(ω+ kV)2

− (kV + ω)2 + Ω2γ

= 0 , (18)

where Ωγ ≡ |eB / γmc|. In the absence of a magnet ic field this dispersion relat ion can be readily

seen to become

ω2 = k2c2 + γ2ω2p . (19)

Here we recover the solut ion for t ransverse E&M waves from eq.(14). Provided we adopt eq. (18)

as the transverse wave dispersion relat ion associated with a velocity shear surface we can use it to

predict the effect of parallel magnet ic fields on KKHI. We will look at this case later.

3. U nequal densit ies wit h (A ) count er -st reaming and (B ) unequal velocit ies

We now return to the general dispersion relat ion, eq.(8):

(k2c2 + γ2−ω

2p− − ω2)1/ 2(kV− − ω)2[(kV+ − ω)2 − ω2

p+ ]

+ (k2c2 + γ2+ω

2p+ − ω2)1/ 2(kV+ − ω)2[(kV− − ω)2 − ω2

p− ] = 0 .

A : U nequal densit ies wit h count er -st reaming velocit ies

This is the case in Alves et al. (2012) where we set V− = −V+ = −V and recall that ω2p± =

4πn± e2/ γ3± m. I not e before I st ar t t hat t here is a t ypo in t he t ext in A lves et al. j ust

below t heir dispersion relat ion [eq.(1) ] where t heir k ≡ kc/ωp+ should be k ≡ kV/ωp+ .

A ddit ional ly, I wi l l show t hat t he leading t erms in t heir square root s are incor rect ,

i .e., should cont ain Lorent z fact ors when t he count er -st reaming flows are relat iv ist ic.

With counter-streaming equal velocit ies but unequal densit ies we can write the dispersion relat ion

as

(k2c2 + γ2ω2p− − ω2)1/ 2(ω+ kV )2[(ω− kV )2 − ω2

p+ ]

+ (k2c2 + γ2ω2p+ − ω2)1/ 2(ω− kV)2[(ω+ kV)2 − ω2

p− ] = 0 , (20)

and this can be rewrit ten as

(k2c2 + γ2ω2p− − ω2)1/ 2[(ω2 − k2V 2)2 − (ω+ kV )2ω2

p+ ]

+ (k2c2 + γ2ω2p+ − ω2)1/ 2[(ω2 − k2V 2)2 − (ω− kV)2ω2

p− ] = 0 . (21)

St udy of t he relat ivist ic velocity shear int er face K K HI inst abilit y

1. T he general dispersion relat ion

I will use init ially use subscripts of “+ ” and “ -” to indicate a “ jet” at x > 0 and ambient at

x < 0 with flow in the y direct ion with a velocity shear surface at x = 0. Here we are infinite in

the z-direct ion.

I begin with the Eigenmode equation from Gruzinov but now generalized to allow complex

frequencies, allow different number densit ies and flow velocit ies on either side of the contact dis-

continuity, and correct a term that was dimensionally wrong in his denominator, i.e., k → kc. The

general Eigenmode equation is:

− (kV − ω)2 + ω2p

− (kV − ω)2(k2c2 + γ2ω2p − ω

2)Ey =

− (kV − ω)2 + ω2p

− (kV − ω)2Ey , (1)

whereωp ≡ 4πne2/ γ3m, perturbations are of the form ei (ky−ωt) and the wavevector, k, is along the

flow direction, and the prime denotes the derivat ive in the x-direct ion.

Within each medium on either side of the shear surface at x = 0 the Eigenmode equation can

be writ ten as:

− (kV+ − ω)2 + ω2p+

− (kV+ − ω)2(k2c2 + γ2+ω

2p+ − ω2)

Ey+ =− (kV+ − ω)2 + ω2

p+

− (kV+ − ω)2Ey+ for x > 0 , (2)

and

− (kV− − ω)2 + ω2p−

− (kV− − ω)2(k2c2 + γ2−ω

2p− − ω

2)Ey− =

− (kV− − ω)2 + ω2p−

− (kV− − ω)2Ey− for x < 0 , (3)

wherecondit ionsareassumed to beuniform on either sideof theshear surface. Thesetwo equations

provide the behavior of the perturbations on either side of the shear surface, that is for

Ey(x, y, t) = Ey(x)ei (ky−ωt)

we have thatd2Ey+

dx2= (k2c2 + γ2

+ω2p+ − ω2)Ey+ for x > 0 , (4)

andd2Ey−

dx2= (k2c2 + γ2

−ω2p− − ω

2)Ey− for x < 0 . (5)

Solut ions to these equations are given by

Ey(x, y, t) = Ey0e∓A± xei (ky−ωt) ,

with A± = (k2c2 + γ2±ω

2p± − ω2)1/ 2 and Ey(x) = Ey0e− A+ x for x > 0 and Ey(x) = Ey0e+ A− x for

x < 0, i.e., Ey(x) declines exponentially away from the shear surface, and we have that Ey+ (x =

0) = Ey− (x = 0) = Ey0 at the shear surface.

Low-frequency limit (V-=0)

– 5 –

Let us now normalize by ωp+ and defineω = ω/ωp+ and k ≡ kV/ωp+ and write in the form used

by Alves et al. to find

(γ2 n−

n++ k 2/ β2 − ω2)1/ 2 (ω + k )2 − (ω2 − k 2)2

+ (γ2 + k 2/ β2 − ω2)1/ 2 n−

n+(ω − k )2 − (ω2 − k 2)2 = 0 . (22)

I n A lves et al. γ2(n− / n+ ) and γ2 in t he leading square root s were wr it t en as (n− / n+ )

and 1, respect ively, and wil l not give t he cor rect solut ion for t ransverse E& M waves

for equal densit y relat iv ist ic count er -st reaming flows. T heir dispersion relat ion [eq.(1) ]

wil l also not give t he cor rect solut ions for unequal densit y relat iv ist ic count er -st reaming

flows.

B : U nequal densit ies and velocit ies

Here we will specialize to cases with V− ≥ 0 which allow mot ion of the ambient , e.g., the

“ needles in a jet ” or “ jet in a jet ” scenarios allowing for high speed features moving through

an already relat ivist ic ambient flow. In what follows we change the notat ion and set nj t = n+ ,

nam = n− , Vj t = V+ , Vam = V− ≥ 0, γj t = γ+ and γam = γ− . With this notat ional change the

general dispersion relat ion can be writ ten as

(k2c2 + γ2amω

2p,am − ω2)1/ 2(ω− kVam )2[(ω− kVj t )

2 − ω2p,j t ]

+ (k2c2 + γ2j tω

2p,j t − ω2)1/ 2(ω− kVj t )

2[(ω− kVam )2 − ω2p,am ] = 0 . (23)

Analyt ic solut ions are not available except in the low (ω< < ωp and kc < < ωp) and high frequency

(ω> > ωp and kc > > ωp) limits.

T he low frequency l imit

In the low frequency limit the dispersion relat ion can be writ ten as

γamωp,amω2p,j t (ω− kVam )2 + γj tωp,j tω

2p,am (ω− kVj t )

2 ∼ 0 . (24)

which yields the quadrat ic equat ion

(γamωp,j t + γj tωp,am)ω2 − 2(γamωp,j tkVam + γj tωp,amkVj t )ω+ (γamωp,j tk2V 2

am + γj tωp,amk2V 2j t ) ∼ 0 .

(25)

with solut ions given by

ω∼(γamωp,j tkVam + γj tωp,amkVj t )

(γamωp,j t + γj tωp,am)± i

(γamωp,j tγj tωp,am )1/ 2

(γamωp,j t + γj tωp,am)k(Vj t − Vam ), (26)

In eq.(25) the real part gives the phase velocity and the imaginary part gives the temporal growth

rate and direct ly shows the dependence of the growth rate on the velocity difference across the

shear surface. In the case where Vam = 0

ω∼(γj tωp,am /ωp,j t )

(1 + γj tωp,am /ωp,j t )kVj t ± i

(γj tωp,am /ωp,j t )1/ 2

(1 + γj tωp,am /ωp,j t)kVj t . (27)

Here it is easy to see that the phase velocity increases and the temporal growth rate decreases as

γj tωp,am /ωp,j t = γ5/ 2j t nam / nj t increases. Recall that ω2

p,j t = 4πnj t e2/ γ3

j t m.

ei(kx −ωt)

(Nishikawa et al. 2013a,b)

Dispersion relation for longitudinal electron mode

of KKHI

γj = 5 γj = 15

nj = na

Kinetic Kelvin-Helmholtz Instability

1. Static electric field grows due to the charge separation by the

negative and positive current filaments

2. Current filaments at the velocity shear generate magnetic field

transverse to the jet along the velocity shear

3. Jet with high Lorentz factor with core-sheath case generate higher

magnetic field even after saturated in the case counter-streaming

case with moderately relativistic jet

4. Non-relativistic jet generate KKHI quickly and magnetic field grows

faster than the jet with higher Lorentz factor

5. For the jet-sheath case with Lorentz factor 15 the evolution of

KKHI does not change with the mass ratio between 20 and 1836

6. Strong magnetic field affects electron trajectories and create

synchrotron-like (jitter) radiation which is in progress

7. KKHI need to be investigated with shocks (jets need to be injected

to ambient plasma)

24/39 Accelerated particles emit waves at shocks

Schematic GRB from a massive stellar progenitor

(Meszaros, Science 2001)

Prompt emission Polarization ?

Simulation box

25/39

3-D simulation

X

Y

Z

jet front

131×131×4005

grids

(not scaled)

1.2 billion particles

injected at z = 25Δ

with MPI code

ambient plasma

26/39

Collisionless shock

Electric and magnetic fields created self-

consistently by particle dynamics randomize

particles

jet ion

jet electron ambient electron

ambient ion

jet

(Buneman 1993)

¶B / ¶t = -Ñ ´ E

¶E / ¶t = Ñ ´ B - J

dm0g v / dt = q(E + v ´ B)

¶r / ¶t +ÑiJ = 0

27/39

Weibel instability

x evz × Bx

jet

J

J

current filamentation

generated

magnetic fields

Time:

τ = γsh1/2/ωpe ≈ 21.5

Length:

λ = γth1/2c/ωpe ≈ 9.6Δ

(Medvedev & Loeb, 1999, ApJ)

(electrons)

3-D isosurfaces of z-component of current Jz for narrow jet (γv||=12.57)

electron-ion ambient -Jz (red), +Jz (blue),

magnetic field lines (white)

t = 59.8ωe-1

Particle acceleration due to the local

reconnections during merging current

filaments at the nonlinear stage

thin filaments merged filaments

32/39

Shock velocity and bulk velocity

trailing edge

leading shock

(forward shock)

contact discontinuity (CD)

jet electrons

ambient electrons

total electrons

Fermi acceleration ?

leading edge

trailing shock

(a) electron density and (b) electromagnetic

field energy (εB, εE) divided by the total

kinetic energy at t = 3250ωpe-1

vcd=0.76c

jet

ambient

vrs=0.56c

total

εE

εB

(Nishikawa et al. ApJ, 698, L10, 2009)

(c)

Shock formation, forward shock, reverse shock

(a) electron density and (b) electromagnetic

field energy (εB, εE) divided by the total

kinetic energy at t = 3250ωpe-1

vcd=0.76c

jet

ambient

vrs=0.56c

total

εE

εB

(Nishikawa et al. ApJ, 698, L10, 2009)

vjf=0.996c

(c)

Shock formation, forward shock, reverse shock

(a) electron density and

(b) electromagnetic

field energy (εB, εE)

divided by the total

kinetic energy at

t = 3250ωpe-1

vcd=0.76c

jet

ambient

vrs=0.56c

total

εE εB

(Nishikawa et al. ApJ, 698, L10, 2009)

vjf=0.996c (c)

Time evolution of the total electron density.

The velocity of jet front is nearly c, the predicted

contact discontinuity speed is 0.76c, and the

velocity of trailing shock is 0.56c.

Time evolution of the total electron density.

The velocity of jet front is nearly c, the predicted

contact discontinuity speed is 0.76c, and the

velocity of trailing shock is 0.56c.

Time evolution of the total electron

density. The velocity of the jet front is ~c,

the predicted contact discontinuity speed

is 0.76c, and the velocity of the reverse

shock is 0.56c.

•Fermi acceleration (Monte Carlo simulations are not self-

consistent; particles are crossing the shock surface many

times and remain accelerated, the strengths of turbulent

magnetic fields are assumed), Some simulations exhibit

Fermi acceleration (Spitkovsky 2008)

•The strength of magnetic fields is estimated based on

equipartition - magnetic field energy is comparable to the

thermal energy): εB ~ u(T)

•The distribution of accelerated electrons is approximated

by the power law (F(γ) = γ−p; p = 2.2?) (εe)

•Synchrotron emission is calculated based on p and εB

•There are many assumptions in this calculation!

Present theory of Synchrotron radiation

39/39

Synchrotron Emission: radiation from accelerated

adapted by

S. Kobayashi

•Electrons are accelerated by the electromagnetic field

generated by the Weibel instability and KKHI (without

the assumption used in test-particle simulations for

Fermi acceleration)

•Radiation is calculated using the particle trajectory in

the self-consistent turbulent magnetic field

•This calculation includes Jitter radiation (Medvedev

2000, 2006) which is different from standard

synchrotron emission

•Radiation from electrons in our simulation is reported

in Nishikawa et al. Adv. Sci. Rev, 47, 1434, 2011.

Self-consistent calculation of radiation

41/39

Radiation from particles in collisionless shock

New approach: Calculate radiation

from integrating position, velocity,

and acceleration of ensemble of

particles (electrons and positrons)

Hededal, Thesis 2005 (astro-ph/0506559)

Nishikawa et al. 2008 (astro-ph/0802.2558)

Sironi & Spitkovsky, 2009, ApJ

Martins et al. 2009, Proc. of SPIE Vol. 7359

Frederiksen et al. 2010, ApJL

43/39

Synchrotron radiation from gyrating electrons in a uniform magnetic field

β

β

n

n

B

β

β

electron trajectories radiation electric field observed at long distance

spectra with different viewing angles

0°

theoretical synchrotron spectrum

1° 2° 3°

6°

time evolution of three frequencies

4°

5°

observer

f/ωpe = 8.5, 74.8, 654.

44/39

Synchrotron radiation from propagating electrons in a uniform magnetic field

electron trajectories radiation electric field observed at long distance

spectra with different viewing angles (helical)

observer

B

gyrating

θ

θγ = 4.25°

Nishikawa et al. astro-ph/0809.5067

45/39

Synchrotron vs. `Jitter’

• (a) Synchrotron emission assumes large-scale

homogeneous magnetic fields

• (b) `Jitter’ radiation (Medvedev 2000) occurs where

the gyro-radius is larger than the randomness of

turbulent magnetic fields

46/39 (Nishikawa et al. astro-ph/0809.5067)

49/39

Radiation from electrons by tracing trajectories self-consistently

using a small simulation system initial setup for jitter radiation

select electrons

randomly (12,150)

in jet and ambient

50/39

final condition for radiation

15,000 steps

dt = 0.005

nω = 100

nθ = 2

Δxjet = 75Δ

tr = 75 w pe

-1

w pe

-1

75Δ

51/39

Calculated spectra for jet electrons and ambient electrons

θ = 0° and 5°

Case D

γ = 15

γ = 7.11

Nishikawa et al. 2009 (arXiv:0906.5018)

Bremesstrahlung (ballistic)

qg = 3.81°

high frequency

due to turbulent

magnetic field

a = λe|δB|/mc2 < 1 (λ: the length scale and |δB| is the magnitude of the fluctuations)

52/39

Hededal & Nordlund (astro-ph/0511662)

3D jitter radiation (diffusive synchrotron radiation) with a ensemble of

mono-energetic electrons (γ = 3) in turbulent magnetic fields

(Medvedev 2000; 2006, Fleishman 2006) (ballistic)

μ= -2

0

2

2d slice of

magnetic field 3D jitter radiation

with γ = 3 electrons

53/39

Dependence on Lorentz factors of jets

θ = 0°

5°

θ = 0°

5°

γ = 15

γ = 100

qg = 0.57

qg = 3.81°

Narrow beaming

angle

54/39

Observations and numerical spectrum

a ≈ 1

Abdo et al. 2009, Science

a b

c d e

Nishikawa et al. 2010

GRB 080916C

a ≈ 1

56/39

System size: 8000 × 240 × 240

Electron-positron: γ = 15

Radiation in a larger system at early time

(b) 200 ≤ t ≤ 275 w pe

-1 w pe

-1

Sampled particles 115,200

(a)150 ≤ t ≤ 225 w pe

-1 w pe

-1

Nishikawa et al. in progress

Summary for synthetic spectra

• Synthetic spectra provide self-consistent from

electrons in turbulent magnetic fields

• In order to compare with observational spectra

further investigation is necessary

• Kinetic Kelvin-Helmholtz instability is also need

to be included for obtaining synthetic spectra

• We need to implement more realistic GRB jet

conditions with Lorentz factor, density,

temperature, etc.

58/39

GRB progenitor (collapsar, merger, magnetar)

EM

emission

relativistic jet Fushin

Raishin

(Tanyu Kano 1657)

(shocks, acceleration)

(god of wind)

(god of lightning)

Gravitational waves